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July 28th, 2017, 02:38 AM   #1
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Fourier Transform

Hello everyone,
am trying to solve this Fourier Trans. problem,
here is the original solution >> https://i.imgur.com/eJJ5FLF.png


Q/ How did he come up with this result and where is my mistake?

All equation are in the above attached picture

here is my attempt,

part 1>> https://i.imgur.com/DT2tJ0y.jpg

part 2>> https://i.imgur.com/jopEoQd.jpg

part 3>> https://i.imgur.com/cKXoekT.jpg
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July 28th, 2017, 03:48 AM   #2
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You are out by a factor of $\omega$ in the final line, and perhaps consequently, a sign change. I suspect it was introduced in part 2 somewhere.
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July 28th, 2017, 04:50 AM   #3
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second attempt

Quote:
Originally Posted by Joppy View Post
You are out by a factor of $\omega$ in the final line, and perhaps consequently, a sign change. I suspect it was introduced in part 2 somewhere.
thanks and here is my second attempt
but also there is a little difference at the final results

https://i.imgur.com/jkVdK7z.jpg

and here is the original solution
https://i.imgur.com/eJJ5FLF.png

can you spot where is the mistake ??
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July 28th, 2017, 05:22 AM   #4
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What did you do on 'line 5'
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July 28th, 2017, 05:29 AM   #5
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Quote:
Originally Posted by Joppy View Post
What did you do on 'line 5'
No, I just write that word to remember it (not that important).
In the previous solution there is (negative sign) which I forgot to include, but now I've included it,
and, as you can see, the solution is somehow still different from the original one.
Can you explain why?

Last edited by skipjack; July 28th, 2017 at 06:50 AM.
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July 28th, 2017, 06:46 AM   #6
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$\displaystyle \int_0^\pi\! e^{-j\omega t}\sin(t)dt = \left[-\frac{e^{-j\omega t}(\cos(t) + j\omega\sin(t))}{1 - \omega^2}\right]_0^\pi = \frac{1 + e^{-j\omega\pi}}{1 - \omega^2}$.

In the "original" solution, $\theta$ should be $t$.
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July 28th, 2017, 07:56 AM   #7
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question

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Originally Posted by skipjack View Post
$\displaystyle \int_0^\pi\! e^{-j\omega t}\sin(t)dt = \left[-\frac{e^{-j\omega t}(\cos(t) + j\omega\sin(t))}{1 - \omega^2}\right]_0^\pi = \frac{1 + e^{-j\omega\pi}}{1 - \omega^2}$.

In the "original" solution, $\theta$ should be $t$.
Thanks indeed, but how did you come up with that?
Can you elaborate more?

And is it correct that he reversed the integrals limit in the original solution?

Last edited by skipjack; July 28th, 2017 at 07:59 AM.
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July 28th, 2017, 08:08 AM   #8
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For that type of integral, see this article.

The limits were swapped to compensate for omitting $j^2$ from the denominators.
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July 28th, 2017, 08:11 AM   #9
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Thanks indeed for your article link, but can you explain more about the results of the integration?

Last edited by skipjack; July 28th, 2017 at 09:13 AM.
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July 28th, 2017, 09:16 AM   #10
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What specific thing are you referring to?
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